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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 5

Lesson 4: Using the first derivative test to find relative (local) extrema

# Finding relative extrema (first derivative test)

Once you find a critical point, how can you tell if it is a minimum, maximum or neither? Created by Sal Khan.

## Want to join the conversation?

• what does local extrema mean?
• Look at the graph of this function. The graph dips down right at the y-axis, which is an important detail, but it isn't really the minimum value of the function since it goes to negative infinity eventually. Therefore, we call that value of the function a local minimum, since it is the smallest value near x=0.

• Is it possible to have two local maximums?
• You could have infinitely many, just look at a sine curve for example.
• Are all critical points maximums/minimums and vice versa? Is this the definition of a critical point (max/min)? Thank you.
• All maximums and minimums are critical points, but it does NOT work the other way around. You can have a critical point that is not a maximum or minimum. In this video the point at x sub 3 is a critical point, but it is NOT a maximum nor minimum. This point is called an inflection point, and future videos explain inflection points.
• Could there be more than 1 local maximums or local minimums per function? How about more than 1 global maximums and global minimums?
• You can only have one value for the absolute max or min, (like f(x)=1) but I think that value can appear more than once, as it does in the sine function. In f(x)=sin(x), I think 1 counts as the absolute max even though it occurs at pi/2, 5pi/2, 9pi/2, etc.
(I think global and absolute maximums are the same thing, but absolute is what I studied.)
• How can we calculate whether the critical number is either maxima or minima without graph?
• Use the second derivative or check f'(x) on either side of the critical number.
• What is an inflection point? Is it a critical point?
• An inflection point is a point on a curve where the curve changes from concave up to concave down or the concave down to concave up. A critical point is where possible maxima values and minima values are.
• This whole video seems useless to me because to see if the surrounding derivative is positive or negative you have to look at the graph, but then you might as well see if its the greatest (or least) point in the area?
• You cannot always tell from the graph. You might get a rough approximation of where the max or min is, but you need the derivative to check the exact value. Also, if you have max and min quite close to each other the graph can look like it has fewer max and min than it really has -- so you need the derivative to be sure. For example, could you tell just by looking at this graph how many maxima and minima there are: http://www.wolframalpha.com/share/clip?f=d41d8cd98f00b204e9800998ecf8427emr9u8s45l4

• In order for a point to be maximum or minimum, must the derivatives approaching from both sides be symmetrical? I mean, does the derivative of points leading up to the critical point from the left have to correspond with its additive inverse on the right?
• No, the derivatives approaching from either side of the maximum or minimum do not have to be symmetrical. Try graphing the function y = x^3 + 2x^2 + .2x. You have a local maximum and minimum in the interval x = -1 to x = about .25. By looking at the graph you can see that the change in slope to the left of the maximum is steeper than to the right of the maximum. And the change in slope to the left of the minimum is less steep than that to the right of the minimum.
• what is the trick in identifying a local minimum or maximum by looking at the graph?